Polar Circle: What Is It About?
A Mathematical Droodle

The orthocenter H of ΔABC serves as the radical center of the circles constructed on the sides of the triangle as diameters. If the feet of the altitudes opposite the vertices A, B, C are denoted Ha, Hb, Hc, then the powers of the point H with respect to the three circles are equal:

(1)

HA·HHa = HA·HHb = HA·HHc.

For an obtuse triangle, the orthocenter lies outside the triangle and outside each of the three circles. (1) then means that the tangents from H to the three circles are equal. The endpoints of those tangents lie on a circle centered at H and perpendicular to all three circles. This circle is known as the polar circle of ΔABC. Thus the polar circle of ΔABC is centered at the orthocenter and have the radius R defined by

(2)

R2 = HA·HHa = HA·HHb = HA·HHc.

The polar circle is only defined for obtuse triangles.

The applet shows a triangle with the altitudes and four (two red and two magenta) circles drawn. There is also a green circle centered at the orthocenter. The radius of the latter is controlled by the scrollbar at the bottom of the applet. One of the red circles is the circumcircle of ΔABC; the second one is its inverse image in the green circle. (Note that the two red circles cross on the green one.) When the radius of the latter is such that the two red circles coincide, you can observe that the circles are orthogonal. In the second pair of circles (drawn in magenta), one is the 9-point circle of ΔABC (the circumcircle of ΔHaHbHc); the second one is its inverse image in the green circle.

The applet helps grasp the following fact: the circumcircle and the 9-point circle of an obtuse triangle are inverse images of each other in the polar circle of the triangle.

The proof is immediate from (2): the inversion in the polar circle exchanges vertices of a triangle with the feet of the altitudes, so that, for example, A is mapped on Ha. Since inversion maps circles onto circle, the circumcircle of ABC is inverted into the circumcircle of HaHbHc. The same fact affords a reformulation: the sides of the triangle are the polars of the opposite vertices with respect to the polar circle. A triangle with this property is called self-conjugate with respect to the circle. (This explains the naming convention.) It follows that every obtuse triangle is self-conjugate with respect to its polar circle.